A375512 -id:A375512 - cp's OEIS frontend

A375580 a(n) is the number of partitions n = x + y + z of positive integers such that xyz is a perfect cube.

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 3, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 2, 2, 2, 2, 1, 2, 3, 2, 2, 3, 2, 0, 1, 1, 3, 1, 3, 2, 2, 1, 1, 1, 2, 1, 4, 1, 2, 3, 3, 3, 3, 1, 1, 4, 2, 2, 2, 3, 1, 2, 3, 1, 3, 4, 1, 3, 2, 2, 1, 2, 2, 3, 3, 2, 4

Offset: 0

Felix Huber, Aug 19 2024

nonn

a(n) is also the number of distinct integer-sided cuboids with total edge length 4*n whose unit cubes can be grouped to a cube.

Conjecture: for n > 176, a(n) > 0. - Charles R Greathouse IV, Aug 20 2024

			a(21) = 3 because the three partitions [1, 4, 16], [3, 6, 12], [7, 7, 7] satisfy the conditions: 1 + 4 + 16 = 21 and 1*4*16 = 4^3, 3 + 6 + 12 = 21 and 3*6*12 = 6^3, 7 + 7 + 7 = 21 and 7*7*7 = 7^3.
See also linked Maple code.

Felix Huber and Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (up to 1000 from Huber)
Felix Huber, Maple codes
David A. Corneth, PARI program

Cf. A338939, A375512, A375576, A103277, A069905.

Maple
```
# See Huber link.
```

a(n)=sum(x=1,n\3, sum(y=x,(n-x)\2, ispower(x*y*(n-x-y),3))) \\ Charles R Greathouse IV, Aug 20 2024

PARI
```
\\ See Corneth link
```

from sympy import integer_nthroot
def A375580(n): return sum(1 for x in range(n//3) for y in range(x,n-x-1>>1) if integer_nthroot((n-x-y-2)*(x+1)*(y+1),3)[1]) # Chai Wah Wu, Aug 21 2024

Trivial upper bound: a(n) <= A069905(n). - Charles R Greathouse IV, Aug 23 2024

A375576 a(n) is the number of partitions n = x + y + z of positive integers such that xyz is a perfect square.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 4, 1, 2, 3, 2, 2, 4, 2, 4, 3, 4, 2, 5, 4, 2, 6, 5, 2, 8, 4, 8, 4, 4, 5, 10, 5, 3, 8, 7, 6, 12, 5, 6, 7, 6, 7, 11, 5, 6, 8, 12, 6, 11, 8, 11, 11, 6, 3, 22, 6, 12, 12, 8, 9, 13, 12, 7, 14, 14, 6, 18, 7, 7, 18, 13, 14, 13, 7, 19, 10

Offset: 0

Felix Huber, Aug 19 2024

nonn

a(n) is also the number of distinct integer-sided cuboids with total edge length 4*n whose unit cubes can be grouped to a square cuboid with height 1.

			a(24) = 4 because the four partitions [2, 4, 18], [3, 9, 12], [4, 4, 16], [4, 10, 10] satisfy the conditions: 2 + 4 + 18 = 24 and 2*4*18 = 12^2, 3 + 9 + 12 = 24 and 3*9*12 = 18^2, 4 + 4 + 16 = 24 and 4*4*16 = 16^2, 4 + 10 + 10 = 24 and 4*10*10 = 20^2.
See also linked Maple code.

Felix Huber, Table of n, a(n) for n = 0..1000
Felix Huber, Maple Codes

Cf. A338939, A375512, A375580.

Maple
```
See Huber link.
```

from sympy.ntheory.primetest import is_square
def A375567(n): return sum(1 for x in range(n//3) for y in range(x,n-x-1>>1) if is_square((n-x-y-2)*(x+1)*(y+1))) # Chai Wah Wu, Aug 22 2024

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 3, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 0, 5, 1, 1, 2, 3, 3, 2, 1, 1, 3, 6, 1, 4, 2, 7, 4, 4, 0, 3, 5, 3, 4, 2, 1, 7, 2, 1, 5, 9, 3, 5, 3, 4, 1, 9, 2, 6, 3, 5, 6, 5, 4, 7, 5, 1, 5, 6, 3, 13, 7, 8, 4, 6, 0, 4, 4, 11, 5, 13, 2

Offset: 1

Felix Huber, Apr 04 2025

nonn

a(n) is the number of distinct cuboids with edge length 4*n whose surface area is half of a square.

Conjecture: a(k) = 0 iff k is an element of {2, 4, 8, 13} union A000244 union A005030.

			The a(14) = 3 partitions [x, y, z] are [1, 1, 12], [1, 4, 9] and [4, 4, 6] because 1*1 + 1*12 + 1*12 = 5^2, 1*4 + 4*9 + 1*9 = 7^2 and 4*4 + 4*6 + 4*6 = 8^2.

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Maple program to calculate the partitions

Cf. A000244, A005030, A066955, A069905, A338939, A375512, A375576, A375580, A375731.

A382407:=proc(n)
    local a,x,y,z;
    a:=0;
    for x to n/3 do
        for y from x to (n-x)/2 do
            z:=n-x-y;
            if issqr(x*y+x*z+y*z) then
                a:=a+1
            fi
        od
    od;
    return a
end proc;
seq(A382407(n),n=1..87);

cp's OEIS Frontend

A375580 a(n) is the number of partitions n = x + y + z of positive integers such that xyz is a perfect cube.

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A375576 a(n) is the number of partitions n = x + y + z of positive integers such that xyz is a perfect square.

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A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.

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Maple

cp's OEIS Frontend

A375580 a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

PARI

Python

Formula

A375576 a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that x*y + y*z + x*z is a perfect square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A375580 a(n) is the number of partitions n = x + y + z of positive integers such that xyz is a perfect cube.

A375576 a(n) is the number of partitions n = x + y + z of positive integers such that xyz is a perfect square.

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.